Characterization of the Fibonacci Cobweb Poset as oDAG

The characterization of Fibonacci Cobweb poset P as DAG and oDAG is given. The dim 2 poset such that its Hasse diagram coincide with digraf of P is constructed. 1 Fibonacci cobweb poset The Fibonacci cobweb poset P has been invented by A.K.Kwaśniewski in [1, 2, 3] for the purpose of finding combinatorial interpretation of fibonomial coefficients and eventually their reccurence relation. In [1] A. K. Kwaśniewski defined cobweb poset P as infinite labeled digraph oriented upwards as follows: Let us label vertices of P by pairs of coordinates: 〈i, j〉 ∈ N0 × N0, where the second coordinate is the number of level in which the element of P lies (here it is the j-th level) and the first one is the number of this element in his level (from left to the right), here i. Following [1] we shall refer to Φs as to the set of vertices (elements) of the s-th level, i.e.: Φs = {〈j, s〉, 1 ≤ j ≤ Fs} , s ∈ N ∪ {0}, where {Fn}n≥0 stands for Fibonacci sequence. Then P is a labeled graph P = (V,E) where V = ⋃ p≥0 Φp, E = {〈 〈j, p〉, 〈q, p + 1〉 〉} , 1 ≤ j ≤ Fp, 1 ≤ q ≤ Fp+1.